Quelles relaxations continues pour le critère l 2 − l 0 ?

For more than two decades, several continuous (and generally separable) penalties approximating (relaxing) the `0-pseudo norm have been proposed. Although some “good” properties for such penalties have been highlighted, the choice of one relaxation rather than another one remains unclear. One approach to compare them is to investigate their fidelity to the initial problem. In other words, do they preserve global minimizers of the initial criteria without adding new local ones? Within the context of the `0 penalized least squares, we have recently studied this question resulting in a class of penalties said exact. In this communication, we present these results and complete them with a study concerning the local minimizers eliminated by such relaxations. In particular, we show that the CEL0 penalty is the one removing the largest number of local minimizers.